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GATE 2025 MT – Question 41

Engineering Mathematics · Linear algebra and matrices · 2 marks · Multiple choice

Which one of the following matrices is orthogonal?

  1. $\begin{bmatrix}1/2&-\sqrt3/2\\-\sqrt3/2&1/2\end{bmatrix}$
  2. $\begin{bmatrix}1/2&-\sqrt3/2\\\sqrt3/2&1/2\end{bmatrix}$
  3. $\begin{bmatrix}1/\sqrt2&-\sqrt3/2\\-\sqrt3/2&1/2\end{bmatrix}$
  4. $\begin{bmatrix}1/\sqrt2&-\sqrt3/2\\\sqrt3/2&-1/\sqrt2\end{bmatrix}$

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Correct answer: (B) $\begin{bmatrix}1/2&-\sqrt3/2\\\sqrt3/2&1/2\end{bmatrix}$

Explanation

An orthogonal matrix has mutually perpendicular unit columns, equivalently AᵀA=I. In B each column has squared norm 1/4+3/4=1 and their dot product is −√3/4+√3/4=0. It is a rotation matrix. Therefore **B**.